Algorithms for Worst-Case Design and Applications to Risk Management [Course Book ed.] 9781400825110

Table of contents :
Contents
Preface
Chapter 1. Introduction to Minimax
Chapter 2. A Survey Of Continuous Minimax Algorithms
Chapter 3. Algorithms For Computing Saddle Points
Chapter 4. A Quasi-Newton Algorithm For Continuous Minimax
Chapter 5. Numerical Experiments With Continuous Minimax Algorithms
Chapter 6 Minimax As A Robust Strategy For Discrete Rival Scenarios
Chapter 7 Discrete Minimax Algorithm For Equality And Inequality Constrained Models
Chapter 8. A Continuous Minimax Strategy For Options Hedging
Chapter 9. Minimax and Asset Allocation Problems
Chapter 10. Asset/Liability Management Under Uncertainty
Chapter 11 Robust Currency Management
Index

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Algorithms for Worst-case Design and Applications to Risk Management

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Algorithms for Worst-case Design and Applications to Risk Management

Berc¸ Rustem Department of Computing Imperial College of Science, Technology & Medicine 180 Queen’s Gate, London SW7 2BZ, UK

Melendres Howe Imperial College and Asian Development Bank 6 ADB Avenue, Mandaluyong City 0401 MM PO Box 789, 0980 Manila, Philippines

PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD

Copyright q 2002 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 3 Market Place, Woodstock, Oxfordshire OX20 1SY All Rights Reserved Library of Congress Cataloging-in-Publication Data applied for. Rustem, Berc¸ and Howe, Melendres Algorithms for Worst-case Design and Applications to Risk Management / Berc¸ Rustem and Melendres Howe p. cm. Includes bibliographical references and index. ISBN 0-691-09154-4 (alk. paper) British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. This book has been composed in Times and Abadi Printed on acid-free paper. 1 www.pup.princeton.edu Printed in the United States of America 10 9 8 7 6 5 4 3 2 1

The gods to-day stand friendly, that we may, Lovers in peace, lead on our days to age! But, since the affairs of men rest still incertain Let’s reason with the worst that may befall. William Shakespeare Julius Caesar, Act 5 Scene 1. Dedicated to those who have suffered the worst case.

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Contents

xiii

Preface Chapter 1. Introduction to minimax 1

2 3 4

Background and Notation 1.1 Linear Independence 1.2 Tangent Cone, Normal Cone and Epigraph 1.3 Subgradiemts and Subdifferentials of Convex Functions Continuous Minimax Optimality Conditions and Robustness of Minimax 3.1 The Haar Condition Saddle Points and Saddle Point Conditions References Comments and Notes

Chapter 2. A survey of continuous minimax algorithms 1 2 3 4

Introduction The Algorithm of Chaney The Algorithm of Panin The Algorithm of Kiwiel References Comments and Notes

Chapter 3. Algorithms for computing saddle points 1

2

3 4 5

Computation of Saddle Points 1.1 Saddle Point Equilibria 1.2 Solution of Systems of Equations The Algorithms 2.1 A Gradient-based Algorithm for Unconstrained Saddle Points 2.2 Quadratic Approximation Algorithm for Constrained Minimax Saddle Points 2.3 Interior Point Saddle Point Algorithm for Constrained Problems 2.4 Quasi-Newton Algorithm for Nonlinear Systems Global Convergence of Newton-type Algorithms Achievement of Unit Stepsizes and Superlinear Convergence Concluding Remarks References Comments and Notes

1 1 5 7 7 10 11 13 15 17 18 23 23 25 30 31 33 34 37 37 37 40 42 42 44 45 49 50 54 58 58 59

viii

CONTENTS

Chapter 4. A quasi-Newton algorithm for continuous minimax 1 2 3 4 5

Introduction Basic Concepts and Definitions The quasi-Newton Algorithm Basic Convergence Results Global Convergence and Local Convergence Rates References Appendix A: Implementation Issues Appendix B: Motivation for the Search Direction d Comments and Notes

Chapter 5. Numerical experiments with continuous minimax algorithms 1 2

3

4 5

Introduction The Algorithms 2.1 Kiwiel’s Algorithm 2.2 Quasi-Newton Methods Implementation 3.1 Terminology 3.2 The Stopping Criterion 3.3 Evaluation of the Direction of Descent Test Problems Summary of the Results 5.1 Iterations when k7x f ðxk ; yÞ; dl $ 2j is Satisfied 5.2 Calculation of Minimum-norm Subgradient 5.3 Superlinear Convergence 5.4 Termination Criterion and Accuracy of the Solution References

Chapter 6. Minimax as a robust strategy for discrete rival scenarios 1 2 3

4

Introduction to Rival Models and Forecast Scenarios The Discrete Minimax Problem The Robust Character of the Discrete Minimax Strategy 3.1 Naive Minimax 3.2 Robustness of the Minimax Strategy 3.3 An Example Augmented Lagrangians and Convexification of Discrete Minimax References

Chapter 7. Discrete minimax algorithm for nonlinear equality and inequality constrained models 1 2 3

Introduction Basic Concepts The Discrete Minimax Algorithm 3.1 Inequality Constraints 3.2 Quadratic Programming Subproblem

63 63 66 70 76 81 86 87 90 91 93 93 94 94 95 96 96 97 97 98 110 110 111 111 112 119 121 121 123 125 125 126 128 132 137

139 139 141 142 142 143

CONTENTS

4 5 6 7

3.3 Stepsize Strategy 3.4 The Algorithm 3.5 Basic Properties Convergence of the Algorithm Achievement of Unit Stepsizes Superlinear Convergence Rates of the Algorithm The Algorithm for Only Linear Constraints References

Chapter 8. A continuous minimax strategy for options hedging 1 2 3 4

Introduction Options and the Hedging Problem The Black and Scholes Option Pricing Model and Delta Hedging Minimax Hedging Strategy 4.1 Minimax Problem Formulation 4.2 The Worst-case Scenario 4.3 The Hedging Error 4.4 The Objective Function 4.5 The Minimax Hedging Error 4.6 Transaction Costs 4.7 The Variants of the Minimax Hedging Strategy 4.8 The Minimax Solution 5 Simulation 5.1 Generation of Simulation Data 5.2 Setting Up and Winding Down the Hedge 5.3 Summary of Simulation Results 6 Illustrative Hedging Problem: A Limited Empirical Study 6.1 From Set-up to Wind-down 6.2 The Hedging Strategies Applied to 30 Options: Summary of Results 7 Multiperiod Minimax Hedging Strategies 7.1 Two-period Minimax Strategy 7.2 Variable Minimax Strategy 8 Simulation Study of the Performance of Different Multiperiod Strategies 8.1 The Simulation Structure 8.2 Results of the Simulation Study 8.3 Rank Ordering 9 CAPM-based Minimax Hedging Strategy 9.1 The Capital Asset Pricing Model 9.2 The CAPM-based Minimax Problem Formulation 9.3 The Objective Function 9.4 The Worst-case Scenario 10 Simulation Study of the Performance of CAPM Minimax 10.1 Generation of Simulation Data 10.2 Summary of Simulation Results 10.3 Rank Ordering 11 The Beta of the Hedge Portfolio for CAPM Minimax

ix 144 145 147 152 156 162 172 176 179 179 181 183 187 187 188 189 190 192 193 194 194 196 196 198 198 204 204 205 207 207 211 213 213 214 214 215 217 218 219 221 222 222 223 224 226

x

CONTENTS

12 Hedging Bond Options 12.1 European Bond Options 12.2 American Bond Options 13 Concluding Remarks References Appendix A: Weighting Hedge Recommendations, Variant B* Appendix B: Numerical Examples Comments and Notes Chapter 9. Minimax and asset allocation problems 1 2

3

4

5

6 7 8

Introduction Models for Asset Allocation Based on Minimax 2.1 Model 1: Rival Return Scenarios with Fixed Risk 2.2 Model 2: Rival Return with Risk Scenarios 2.3 Model 3: Rival Return Scenarios with Independent Rival Risk Scenarios 2.4 Model 4: Fixed Return with Rival Benchmark Risk Scenarios 2.5 Efficiency Minimax Bond Portfolio Selection 3.1 The Single Model Problem 3.2 Application: Two Asset Allocations Using Different Models 3.3 Two-model Problem 3.4 Application: Simultaneous Optimization across Two Models 3.5 Backtesting the Performance of a Portfolio on the Minimax Frontier Dual Benchmarking 4.1 Single Benchmark Tracking 4.2 Application: Tracking a Global Benchmark against Tracking LIBOR 4.3 Dual Benchmark Tracking 4.4 Application: Simultaneously Tracking the Global Benchmark and LIBOR 4.5 Performance of a Portfolio on the Dual Frontier Other Minimax Strategies for Asset Allocation 5.1 Threshold Returns and Downside Risk 5.2 Further Minimax Index Tracking and Range Forecasts Multistage Minimax Portfolio Selection Portfolio Management Using Minimax and Options Concluding Remarks References Comments and Notes

Chapter 10. Asset/liability management under uncertainty 1 2

Introduction The Immunization Framework 2.1 Interest Rates 2.2 The Formulation

226 226 229 233 235 236 237 244 247 247 249 250 250 251 251 252 252 253 254 256 257 258 261 261 264 266 267 269 271 271 273 277 284 288 289 290

291 291 296 296 296

CONTENTS

3 4 5 6

7 8

9

Illustration The Asset/Liability (A/L) Risk in Immunization The Continuous Minimax Directional Immunization Other Immunization Strategies 6.1 Univariate Duration Model 6.2 Univariate Convexity Model The Stochastic ALM Model 1 The Stochastic ALM Model 2 8.1 A Dynamic Multistage Recourse Stochastic ALM Model 8.2 The Minimax Formulation of the Stochastic ALM Model 2 8.3 A Practical Single-stage Minimax Formulation Concluding Remarks References Comments and Notes

Chapter 11. Robust currency management 1 2 3 4 5

6 7 8

Index

Introduction Strategic Currency Management 1: Pure Currency Portfolios Strategic Currency Management 2: Currency Overlay A Generic Currency Model for Tactical Management The Minimax Framework 5.1 Single Currency Framework 5.2 Single Currency Framework with Transaction Costs 5.3 Multicurrency Framework 5.4 Multicurrency Framework with Transaction Costs 5.5 Worst-case Scenario 5.6 A Momentum-based Minimax Strategy 5.7 A Risk-controlled Minimax Strategy The Interplay between the Strategic Benchmark and Tactical Management Currency Management Using Minimax and Options Concluding Remarks References Appendix: Currency Forecasting Comments and Notes

xi 300 303 308 309 309 312 315 325 325 330 333 335 335 337 341 341 345 351 357 359 359 362 363 365 367 369 371 373 374 375 376 376 378 381

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Preface

The conventional approach to decisions under uncertainty is based on expected value optimization. The main problem with this concept is that it neglects the worst-case effect of the uncertainty in favor of expected values. While acceptable in numerous instances, decisions based on expected value optimization may often need to be justified in view of the worst-case scenario. This is especially important if the decision to be made can be influenced by such uncertainty that, in the worst case, might have drastic consequences on the system being optimized. On the other hand, given an uncertain effect, some worst-case realizations might be so improbable that dwelling on them might result in unnecessarily pessimistic decisions. Nevertheless, even when decisions based on expected value optimization are to be implemented, the worstcase scenario does provide an appropriate benchmark indicating the risks. This book is intended for the dual role of proposing worst-case design for robust decisions and methods and algorithms for computing the solution to quantitative decision models. Actually, very little space is devoted to justify worst-case design. This is implicit in the optimality condition of minimax, discussed in Chapter 1. In Chapter 6, the robustness of worst-case optimal strategies are considered for discrete scenarios. Subsequent chapters illustrate the property. Basically, the performance of the minimax optimal strategy is noninferior for any scenario, and better for those other than the worst case. As such, worst-case design needs no further justification as a robust strategy than a deterministic optimal strategy requires in view of suboptimal alternatives. In the book, we consider methods for optimal decisions which take account of the worst-case eventuality of uncertain events. The robust character of minimax, mentioned above, is central to the usefulness of the strategies discussed in this book. The discrete minimax strategy ensures a guaranteed optimal performance in view of the worst case and this is assured for all scenarios: if any scenario, other than the one corresponding to the worst case is realized, performance is assured to improve. The continuous minimax strategy provides a guaranteed optimal performance in view of a continuum of scenarios. If this continuum is taken as scenarios varying between upper and lower bounds, performance is assured over the worst case defined between upper and lower bounds. As such, continuous minimax is a forecaster’s dream as it provides the opportunity for specifying forecasts defined over a range,

xiv

PREFACE

rather than point forecasts. Despite all this, however, we stress that these are merely computational tools. If the forecaster tries to specify too many discrete forecasts, in an attempt to cover most possibilities, discrete minimax may yield too pessimistic strategies or even run into numerical, or computational, problems due to the resulting numerous scenarios. Similarly, as the upper and lower bounds on a range forecast get wider, to provide coverage to a wider set of possibilities, the minimax strategy may become pessimistic. Thus, scenarios have to be chosen with care, among genuinely likely values. The minimax strategy will then answer the legitimate question of what the best strategy should be, in view of the worst-case. The stochastic characterization of uncertainty relies on the average or expected performance of the system in the presence of uncertain effects. The book provides the means for taking account of the effect of the worst case which would, in general, not be reflected in average or expected performance evaluation of the system. While expected performance optimization is often adequate, it is the realization of the worst case that mostly causes the failure of systems. Hence, all decisions need to take account of the worst case and the expected performance of the system. Through its inherent pessimism, the minimax strategy may lead to a serious deterioration of performance. Alternatively, the realization of the worst-case scenario may result in an unacceptable performance deterioration for the strategy based on expected value optimization. Neither minimax nor expected value optimization provide a substitute to wisdom. At best, they can be regarded as risk management tools for analyzing the effects of uncertain events. The book is intended for graduate students, researchers in minimax and for practitioners of risk management in economics, engineering design, finance, management science, operations research. After an introductory Chapter 1 in which the basic concepts and fundamental results used later in the book are discussed, we have devoted Chapters 2–7 to algorithms and Chapters 8–11 to risk management applications in finance. Specifically, we survey continuous minimax algorithms in Chapter 2 and discuss the solution of a subclass, the saddle point problem, in Chapter 3. A quasi-Newton algorithm for continuous minimax is developed in Chapter 4 which has desirable convergence properties and has proved to be successful in practical applications. The latter is considered in Chapter 5 particularly from the point of view of justifying the practical use of a simplified version of the algorithm in Chapter 4. The discrete minimax problem is introduced in Chapter 6 and a quasi-Newton algorithm for the nonlinearly constrained problem is developed in Chapter 7. In Chapter 8, the application of continuous minimax to options and hedging is discussed. The application of mainly the discrete minimax problem to portfolio optimization is explored in Chapter 9. The worst-case analysis of the asset-liability management problem and exchange rate scenarios are considered in Chapters 10 and 11, respectively.

PREFACE

xv

The focus of the discussion is static optimization. Specific problems of dynamical systems are omitted. For linear dynamical systems, the interested reader might wish to consult the H 1 control literature. Among other items omitted, we are acutely aware of the absence of engineering applications. Worst-case design in engineering is perhaps the most intuitively obvious area in which minimax can yield tangible benefits. This is not because of lack of trying to get several members of the UK manufacturing sector interested, rather the reluctance of some of those contacted to embark on new concepts and developments. Fortunately, EPSRC has recognized the potential of the area by funding a new research project on the subject. The results of this will be reported in future research papers. Throughout the development of the risk management concepts in finance, the authors have benefited from extensive discussions with Michael Selby. We are indebted to Marc Hendriks for strong support from a practitioner’s point of view and to Rudi Bogni for initially validating the usefulness of the approach. Last, but by no means least, Robin Becker has provided highly useful critical evaluations of worst-case design. Berc¸ Rustem ([email protected]) Melendres Howe ([email protected])

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Chapter 1 Introduction to minimax

We consider the problem of minimizing a nondifferentiable function, defined by the maximum of an inner function. We refer to this objective function as the max-function. In practical applications of minimax, the max-function takes the form of a maximized error, or disutility, function. For example, portfolio selection models in finance can be formulated in a scenario-based framework where the max-function takes the form of a maximized risk measure across all given scenarios. To solve the minimax problem, algorithms requiring derivative information cannot be used directly and the usual methods that do not require gradients are inadequate for this purpose. Instead of gradients, we need to consider generalized gradients or subgradients to formulate smooth methods for nonsmooth problems. The minimax notation is introduced with relevant concepts in convex analysis and nonsmooth optimization. We consider the basic theory of continuous minimax, characterized by continuous values of maximizing and minimizing variables, and associated optimality conditions. These need to be satisfied at the solution generated by all algorithms. The problem of discrete minimax, with continuous minimization but discrete maximization variables, and related conditions are considered in Chapters 6 and 7. 1 BACKGROUND AND NOTATION Equation and section numbering follow the following rule: (1.2.3) refers to Equation 3 in Chapter 1, Section 2. In Chapter 1 only, this is referred to as (2.3), elsewhere as (1.2.3). Chapter 1, Section 2 is referred to in Chapter 1 only as Section 2, elsewhere as Section 1.2. In this book, we consider strategies, algorithms, properties and applications of worst-case design problems. When taking decisions under uncertainty, it is desirable to evaluate the best policy in view of the worst-case uncertain effect. Essentially, this entails minimax formulations in which the best decision and the worst case is determined simultaneously. In this sense, optimality is defined over all possible values of the uncertain effects as opposed to certain likely realizations. Worst-case design is useful in all disciplines with rival representations of the same system. For example, in economics Chow

2

CHAPTER 1

(1979) and Becker et al. (1986) consider rival macroeconomic models of the same economy. A similar approach to resource allocation is discussed in Pang and Yu (1989). The robustness property is explored in Hansen et al. (1998). In finance, Howe et al. (1994, 1996), Dert and Oldenkamp (1997), Howe and Rustem (1997), Ibanez (1998) and Rustem et al. (2000) consider worst-case decisions in options pricing and portfolio optimization. In engineering, BenTal and Nemirovski (1993, 1994) discuss truss topology design under rival load scenarios. In control systems, H 1 -control theory is essentially an equivalent minimax formulation for the uncertainties in the system and this aspect is explored in Bas¸ar and Bernhard (1991). Rival representations can be characterized either in terms of a discrete choice, such as two or more models of the same system, or as values from a continuous range, such as all the values an uncertain variable may take within an upper and lower bound. The worst-case design or minimax problem can thus be formulated as min max f ðx; yÞ:

x[Rn y[Y

ð1:1Þ

where x [ R n is a column vector of real numbers, denoting the decision variables in the n-dimensional Euclidian space. The vector y represents the uncertain variables and is defined over the feasible set Y. An equivalent problem to the above formulation is given by min FðxÞ

x[Rn

where FðxÞ ¼ max f ðx; yÞ y[Y

is the max-function. If Y is a set of continuous variables, then the problem is known as continuous minimax. An example for such a set is n o Y ¼ y [ Rm j y‘i # yi # yUi ; i ¼ 1; …; m where y‘i and yUi are the lower and upper bounds on the ith element of y. Algorithms for this problem are discussed in detail in Chapters 2–5. If Y consists of a discrete set of values, the corresponding problem is known as discrete minimax. We consider equality and inequality constraints on x in particular for the case of discrete min-max. The problem is expressed as n o f i ðxÞ j gðxÞ ¼ 0; hðxÞ # 0 min max x

i[f1;2;…;mg

where f ðxÞ is the value corresponding to the ith member of the discrete set {1; 2; …; m} over which the maximum is evaluated and g; h are vectors of equality and inequality constraints. Properties of, and algorithms for, this formulation are considered in Chapters 6 and 7. i

3

INTRODUCTION TO MINIMAX

In this section, we review some of the basic concepts used elsewhere in the book. Some of the more preliminary material is covered in Comments and Notes (CN 1–CN 11) at the end of the chapter. We start by considering the closed line segment joining x and z, denoted by ½x; z, ½x; z ; fw [ Rn j w ¼ lx 1 ð1 2 lÞz; 0 # l # 1g

ð1:2Þ

and by ðx; zÞ the corresponding open line segment. Convexity is invoked extensively in minimax. We define this property for sets and functions. A set C , Rn is called convex if ½x; z , C for all x; z [ C. The linear combination J X

l j xj

j¼1

is called a convex combination of vectors x1 ; …; xJ [ Rn if

lj $ 0; j ¼ 1; …; J

and

J X

lj ¼ 1:

j¼1

The convex hull of a set C , R , denoted by conv C, is the set of all convex combinations of points in C. Let C be a convex set. A function f : C , Rn ! R1 is said to be convex if n

f ðlx 1 ð1 2 lÞzÞ # lf ðxÞ 1 ð1 2 lÞf ðzÞ

ð1:3Þ

for x; z [ C and l [ ð0; 1Þ. If strict inequality holds for x – z and l [ ð0; 1Þ, then f is strictly convex. A function defined on C is said to be (strictly) concave if the function f ¼ 2g is (strictly) convex. Lemma 1.1 Let f ðxÞ [ C1 (see CN 3). Then, f is convex over a convex set C if and only if f ðzÞ $ f ðxÞ 1 7f ðxÞðz 2 xÞ;

;z; x [ C:

The function is strictly convex if this inequality is strict.

Proof. Let f be convex. Then, we have f ðlx 1 ð1 2 lÞzÞ # lf ðxÞ 1 ð1 2 lÞf ðzÞ;

;l [ ½0; 1

and hence f ðlx 1 ð1 2 lÞzÞ 2 f ðzÞ # f ðxÞ 2 f ðzÞ: l This expression yields the required inequality for l ! 0. Let f ðzÞ $ f ðxÞ 1 7f ðxÞðz 2 xÞ;

;z; x [ C

4

CHAPTER 1

and x ¼ lx1 1 ð1 2 lÞy for some x1 [ C and l [ ½0; 1. We have the inequalities f ðx1 Þ $ f ðxÞ 1 7f ðxÞðx1 2 xÞ f ðyÞ $ f ðxÞ 1 7f ðxÞðy 2 xÞ: Multiplying the first of these inequalities by l and the second by (1 2 l ) and adding yields

lf ðx1 Þ 1 ð1 2 lÞf ðyÞ $ f ðxÞ 1 7f ðxÞðlx1 1 ð1 2 lÞy 2 xÞ: Substituting x ¼ l x1 1 (1 2 l )y yields

l f ðx1 Þ 1 ð1 2 lÞf ðyÞ $ f ðlx1 1 ð1 2 lÞyÞ: A Lemma 1.2 Let f ðxÞ [ C2 . Then, f is convex (strictly convex) over a convex set C containing an interior point if and only if the Hessian matrix 72 f ðxÞ of f is positive semi-definite (positive definite) (see CN 4) throughout C. Proof. Using the second order Taylor expansion of f (see CN 5), we have f ðyÞ ¼ f ðxÞ 1 k7f ðxÞ; y 2 xl 1

1 ky 2 x; 72 f ðx 1 lðy 2 xÞÞðy 2 xÞl 2

for some l [ ½0; 1 (k·,·l is defined in CN 2). If the Hessian is positive semidefinite everywhere, we have f ðyÞ $ f ðxÞ 1 k7f ðxÞ; y 2 xl

ð1:4Þ

and, in view of Lemma 1.1, this establishes the convexity of f. We show that if 72 f ðxÞ is not positive semi-definite for some x [ C, then f is not convex. Let 72 f ðxÞ not be positive semi-definite for some x [ C. By continuity of 72 f ðxÞ it can be assumed, without loss of generality that x is an interior point of C. There is a y [ C such that kx 2 y; 72 f ðxÞðx 2 yÞl , 0: Again, by continuity of the Hessian, y may be selected such that kx 2 y; 72 f ðx 1 lðy 2 xÞÞðx 2 yÞl , 0;

;l [ ½0; 1:

In view of the Taylor expansion above, (1.4) does not hold and by Lemma 1.1, f is not convex. A An important property of convexity of f is the uniqueness of its minimum, as stated in the following result.

5

INTRODUCTION TO MINIMAX

Lemma 1.3 Let f be a convex function defined over the convex set C. Then, any local minimum of f is also a global minimum (see CN 11). Proof. Suppose that x1 is a local minimum. Suppose also that there is another point x2 [ C, x1 – x2 , with f ðx2 Þ , f ðx1 Þ. Then, for ax1 1 ð1 2 aÞx2 , a [ ð0; 1Þ, we have   f a x1 1 ð1 2 aÞx2 # a f ðx1 Þ 1 ð1 2 aÞf ðx2 Þ , f ðx1 Þ which contradicts that x1 is a local minimum point.

A

If f ðlxÞ ¼ lf ðxÞ for all l $ 0, then f is said to be positively homogeneous. If f ðx 1 wÞ # f ðxÞ 1 f ðwÞ;

;x; w [ Rn

then f is subadditive. A positively homogeneous subadditive function is always convex. A convex function f : Rn ! R1 is locally Lipschitz continuous (see CN 8) at x, for any x [ Rn (Makela and Neittaanmaki, 1992, p. 8, Theorem 2.1.1). The concept of linear (in)dependence of a set of vectors is used extensively, in particular for the analysis and characterization of the gradients of the maxfunction, in conjunction with Caratheodory’s Theorem which we also cover in this section. We commence with a definition of linear independence.

1.1 Linear Independence A set of vectors {x1 ; …; xr } in R n is linearly independent if and only if the equality r X

aj x j ¼ 0

ð1:5Þ

j¼1

is achieved for aj ¼ 0, j ¼ 1; …; r. If r $ n 1 1, any vectors x1 ; …; xr in R n are linearly dependent. Hence there exist numbers b1 ; …; br such that r X

jbj j . 0

j¼1

and

r X

bj xj ¼ 0:

ð1:6Þ

j¼1

We note that b j can be positive or nonpositive. If r $ n 1 2, then any vectors x1 ; …; xr in R n are linearly dependent with (1.6) including the additional condition that r X

bj ¼ 0:

j¼1

To verify that (1.7) holds for r ¼ n 1 2, we represent

ð1:7Þ

6

CHAPTER 1 nX 12

bj x j ¼ 0

j¼1

as nX 11

bj xj 1 bn12 xn12 ¼ 0:

ð1:8Þ

j¼1

Since r ¼ n 1 2 . n 1 1, xn12 can be represented as a linear combination of {x1 ; …; xn11 } as nX 11

aj xj ¼ xn12 :

j¼1

We let

aj ¼

bj nX 11

bj

j¼1

which yields nX 11 i¼1

b i xi ¼ xn12 : nX 11 j b j¼1

Returning to (1.8),

9 8 > > > > > > > > i nX 11 11 =

> > i¼1 i¼1 j> > > > b > ; : j¼1

which reduces to nX 11 j¼1

bj 1 bn12 ¼ 0 ¼

nX 12

bj :

j¼1

Thus, (1.7) holds for r ¼ n 1 2. The argument can be straightforwardly extended to r . n 1 2. An equivalent, and shorter, demonstration of (1.7) is done by considering the n 1 1 dimensional vectors h iT xk ¼ xk1 ; …; xkn ; 1 [ Rn11 ; k ¼ ½1; …; r; r $ n 1 2

7

INTRODUCTION TO MINIMAX

where xik is the ith element, i ¼ 1; …; n, of xk . There are numbers b k, Pr k k¼1 jb j . 0, satisfying r X bk x k ¼ 0 k¼1

which is equivalent to (1.7).

1.2 Tangent Cone, Normal Cone and Epigraph The tangent cone of a convex set C, at x [ C, is given by n o ZC ðxÞ ; y [ Rn j ’ti # 0 and yi ! y with ðx 1 ti yi Þ [ C :

ð1:9Þ

The normal cone of the convex set C, at x [ C, is given by Z^ C ðxÞ ; fz [ Rn j ky; zl # 0; ;y [ ZC ðxÞg: The epigraph of a function f : R ! R is n o epi f ; ðx; rÞ [ Rn £ R1 j f ðxÞ # r : n

ð1:10Þ

1

ð1:11Þ

The epigraph of a convex function f : Rn ! R1 is a closed convex set (Makela and Neittaanmaki, 1992, Theorem 2.3.7).

1.3 Subgradients and Subdifferentials of Convex Functions The subdifferential of a convex function f : Rn ! R1 , at x [ Rn , is the set n o 2f ðxÞ ; g [ Rn j f ðx 0 Þ $ f ðxÞ 1 gT ðx 0 2 xÞ; ;x 0 [ Rn : ð1:12Þ Each element g [ 2f ðxÞ is called a subgradient of f at x. The directional derivative in the direction v [ Rn satisfies f 0 ðx; vÞ ¼ lim t#0

f ðx 1 tvÞ 2 f ðxÞ : t

ð1:13Þ

More generally, the Clarke (1983) generalized derivative of a locally Lipschitz function f at x in the direction v [ Rn is defined by f 0 ðx; vÞ ; lim sup y!x t#0

f ðy 1 tvÞ 2 f ðyÞ : t

ð1:14Þ

The following important result is used to define a subgradient of the maxfunction where the maximizer is nonunique, in terms of the subdifferential. Theorem 1.1 (Caratheodory’s Theorem) Let G be a set in the finite dimensional Euclidian space, G # R n. Then, any vector

8

CHAPTER 1

g [ conv G may be expressed as a convex combination of at most n 1 1 vectors in G. Proof. This result is widely known (e.g., Demyanov and Malozemov, 1974, Appendix 2, Lemma 1.1). Its proof does provide some insight for the subsequent discussion. Consider the definition of the convex hull of G ( ) r r X X k k k conv G ¼ x ¼ a xk [ G; xk [ G; a $ 0; a ¼ 1; r ¼ 1; 2; … : k¼1

k¼1

ð1:15Þ Suppose that some x [ G cannot be represented by (1.15) with less than n 1 2 terms (after those with ak ¼ 0 have been discarded), that is, r $ n 1 2 in any representation x¼

r X

ak x k ;

xk [ G;

ak . 0;

k¼1

r X

ak ¼ 1:

ð1:16Þ

k¼1

From Pthe discussion on linear independence, if r $ n 1 2, there exist numbers b k, rk¼1 jbk j . 0, such that r X

bk xk ¼ 0;

k¼1

r X

bk ¼ 0:

ð1:17Þ

k¼1

Let

e ¼ min k

{kjb .0}

ak . 0; bk

ak ¼ ak 2 ebk ;

k [ {1; …; r}:

Vector x can be expressed as x¼

r X

a k xk ;

k¼1

r X

ak ¼ 1:

ð1:18Þ

k¼1

To show the equivalence between (1.16) and (1.18), we expand (1.18) to yield x¼

r X

ðak 2 ebk Þxk ;

k¼1

x¼

r X

r X

ðak 2 ebk Þ ¼ 1

k¼1

ak x k 2 e

k¼1

r X

bk xk ;

k¼1

r X

ak 2 e

k¼1

In view of (1.17), we have x¼

r X k¼1

a k xk ;

r X k¼1

ak ¼ 1:

r X k¼1

bk ¼ 1:

9

INTRODUCTION TO MINIMAX

Considering positive and nonpositive b k separately, we see that ak $ 0, k [ {1; …; r}. At least one ak must vanish and this corresponds to subscript (  k ) a k [ k  k ¼ e : b Thus, x is a convex combination of at most r 2 1 vectors of G. Iterating this procedure sufficiently many times, we reach a representation r # n 1 1, (1.17) is no longer satisfied, and we have the desired result. A We invoke Caratheodory’s Theorem in the context of the subdifferential

2FðxÞ ¼ conv G where

  G ¼ 7x f ðx; yÞ j y ¼ arg max f ðx; yÞ y[Y

and

8
> gai ðFia Þ 1 yai gi ðFi Þ 1 yi > > > C> B x^ui 21 > > C B > a e i;0 i;0 > > C> B i¼1 > > > C B 2aB ! ! !C> > > > > a a a n   = < C B X gj ðFj Þ 1 yj gj ðFj Þ 1 yj @ 1 x^h 1 rjc;base 2 rjc;foreign A 21 min maxn j n u a e > > j;0 j;0 X^ [R y[R > > j¼1 > > > > > > * + > > > > h i h i > > > > u ^h u ^h ^ ^ ^ > > ; : 1ð1 2 aÞ X ; X ; C X ; X

ð3:7Þ subject to

k1; X^ u l 1 k1; X^ h l ¼ 1 x^ui 1 x^hi ¼ wi ;

2

xu1

3

6 7 6 . 7 X^ u ¼ 6 .. 7; 4 5 xun

for all foreign assets i ^ Xu $ 0

X^ h $ 0 2 h3 x1 6 7 6 . 7 7 X^ h ¼ 6 6 .. 7; 4 5 xhn

2

w1

3

6 7 6 7 W ¼ 6 ... 7: 4 5 wn

ROBUST CURRENCY MANAGEMENT

355

Additionally, gi and yi are chosen to ensure the following condition is satisfied: !     1 gij ðFij Þ 1 yij ¼ 1; i – j gi ðFi Þ 1 yi gj ðFj Þ 1 yj The last condition is again the triangulation requirement for currencies, similar to the constraint found in Section 2. The solutions to (3.7) are the optimal allocations to unhedged and hedged assets. The foreign currency exposure is given by the summation of all original allocations W that refer to the same foreign country. The optimal hedge ratio for a particular currency then becomes the ratio of the sum of all hedged allocations to the sum of all original allocations that refer to the same foreign country. The currency hedge ratio is given by P h x^j ð3:8Þ h¼ P wk where j is a counter that refers to synthetic hedged assets of one particular currency and k refers to the actual foreign asset of the same currency. For each currency, (3.8) can be calculated to find the optimal currency hedge ratios. Thus, for each point on the efficient frontier, a set of optimal hedge ratios can be determined. The choice of which point on the efficient frontier, and consequently which set of hedge ratios are relevant, depends on the portfolio manager. Formulation (3.7) is the mean-variance solution to the hedge ratio problem. We present the robust formulation of the currency hedge ratio problem. Recall that the modeling of asset returns and currency returns involve the estimation of an equilibrium value based on a vector of factors with some specific estimation error. By providing ranges for the estimation error, one finds a more robust solution to the hedge ratio problem. In the minimax formulation below, we depart from expected value optimization to the minimization of the worst case. The worst case is a scenario (or a set of scenarios corresponding to multiple maximizers) that lies within some range of the estimation error. The minimax equivalent of (3.7) is given by 19 0 ! ! ! 8 n X > > gai ðFia Þ 1 yai gi ðFi Þ 1 yi > > > C> B ^ x 2 1 i > C> B > > ai;0 ei;0 > > C B i¼1 > > > > C B 2 a ! ! ! > > C B > > a a a n   = < C B X g ðF Þ 1 y g ðF Þ 1 y j j j j j j c;foreign c;base A @ 1 x^ 1 r 2 1 2 r min maxn j j j aj;0 ej;0 > X^ u [Rn y[R > > > j¼1 > n > > X^ h [Rn ya [R > > > * + > > > > h i h i > > > > u h u h ^ ^ ^ ^ ^ > > 1ð1 2 aÞ X ; X ; C X ; X ; :

ð3:9Þ subject to

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CHAPTER 11

ylower # yi # yupper ; i i

i ¼ 1; …; n

ya;lower # yaj # ya;upper ; j j

j ¼ 1; …; n

where n is the number of foreign assets, k1; X^ u l 1 k1; X^ h l ¼ 1 x^ui 1 x^hi ¼ wi ;

for all foreign assets i X^ u $ 0 X^ h $ 0

2

xu1

3

6 7 6 . 7 X^ u ¼ 6 .. 7; 4 5 xun

2

xh1

3

6 7 6 . 7 7 X^ h ¼ 6 6 .. 7; 4 5 xhn

2

w1

3

6 7 6 . 7 W ¼ 6 .. 7: 4 5 wn

The currency forecast error yi associated with each asset i clearly comes from the mcur currency models only. This means that the first constraint of (3.9) on each yi is the result of the constraints imposed by the mcur currency models, that is, # yk # yupper ; ylower k k

k ¼ 1; …; mcur

Additionally, the following condition on the cross currencies of the original mcur currency models must be satisfied: ylower # ykl # yupper ; k ¼ 1; …; mcur 2 1; l ¼ 1; …; mcur 2 1; k – l kl kl k ¼ 1; …; mcur 2 1 l ¼ 1; …; mcur 2 1 (see CN 4), and   gk ðFk Þ 1 yk



  1 gkl ðFkl Þ 1 ykl ¼ 1; gl ðFl Þ 1 yl

k – l:

The currency hedge ratio is again given by (3.8) which is used to compute, for each currency, the optimal hedge ratios. For each point on the efficient frontier, a set of optimal hedge ratios can be determined. Similar to the meanvariance version, the choice of which point on the efficient frontier, and consequently which set of hedge ratios are relevant, depends on the portfolio manager.

ROBUST CURRENCY MANAGEMENT

357

Formulations (3.7) and (3.9) are respectively the mean-variance and the minimax versions that yield the optimal hedge ratio per currency represented in the international portfolio. Either formulation gives the strategic currency benchmark, that is, the currency hedge ratios, that provide a guide to the longterm hedging requirements of the international portfolio. The hedge ratio varies across currencies. For a usd-based investor, a period of dollar strength may result in high hedge ratios, while a period of dollar weakness may result in low hedge ratios. Once the hedge ratios have been established, these represent the strategic hedges that need to be put in place to guard against long-term currency risk. We present the tactical management of currency that deals with short-term currency risk. Sections 4 and 5 consider the tactical re-balancing of the currency exposure with a view to enhancing further the total returns from the international portfolio by seeking to gain short-term returns on the currency components. 4 A GENERIC CURRENCY MODEL FOR TACTICAL MANAGEMENT The management of short-term fluctuations in currencies can add extra currency returns on top of the returns from a long term currency benchmark. In Sections 2 and 3, we construct strategic currency benchmarks that address the long-term issue of currency management and currency hedging. In the present section, we consider a tactical system that addresses short-term fluctuations in currencies. We discuss a generic currency model that produces a tactical signal indicating whether a portfolio should be long or short in a particular currency. Deviating from the usual definition, we adopt a currency model that generates a projection (i.e., a forecast) of an exchange rate in the future. Here, a model not only generates a forecast but also translates this into an implied return and then into an appropriate portfolio re-balancing recommendation. Hereafter, we refer to a currency model as a signal-generating model whose positive signal in a currency means holding a long position in that currency, with the size of the position equal to the size of the signal, and negative signal means holding a short position. The horizon of the model may vary between 3 and 6 months, depending on the dominance of factors within the model. As can be seen in Figure 4.1, switches in the direction of the recommendation happen within this time frame. This suggests that the model attempts to capture the trending nature of the currency. In reality, currency models do not possess a strong enough predictive power to claim consistent out-performance in the foreign exchange markets. In the figure, we see a more typical currency model, superimposed on the

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Figure 4.1

Yen/usd (line) against signals (bar) generated by a model.

nonsmoothed yen/usd. From the figure, we see that the model does not consistently lead the currency. The model’s signal can vary in magnitude from 0% to over 100%, depending on the particular application, but if used in the context of benchmark-tracking, magnitudes in the order of 10% are more typical. Moreover, the signals of the model change over time, perhaps a reflection of the shift in dominance of factors within the model. Such a model may be used in the currency markets if its overall performance yields a sufficiently high return per unit of risk taken. The generic currency model used in this section requires some clarification. Let Mi be a currency model for currency i, producing a signal si;t at time t. The signal may be a complex average of signals from different factors that make up Mi . The particular use of this generic currencycurmodel is for capturing shortterm movements in a currency. Let M [ Rm be a multicurrency model cur comprising mcur currencies, producing a vector of signals St [ Rm , at time t. The interaction between the mcur currencies would become apparent later when we show that any two pairs within this multicurrency framework may produce signals of opposite signs: this implies a simultaneous buying and selling of US dollars as a cross hedge (see CN 2). A signal si;t produced by a currency model would change over time. Let di;t ¼ si;t 2 si;t21 be the change in signal value from time t 2 1 to t. di;t represents a trade recommendation that should be implemented to shift the currency cur holding from si;t21 to si;t . Let Dt [ Rm be the vector of trades at time t. We adopt the US dollar as the base currency, and assume that a currency model will produce signals from the perspective of that currency. In other words, a trade recommendation di;t of, say, 1% for a particular currency means buying 1% of that currency and selling 1% of US dollars. The time series of signals si;t produced by currency model Mi are accompanied by a time series of trades di;t . Depending on the way factors in the model shift their individual signals, di;t can vary in direction several times within the modeling horizon. Recall that di;t is the trade recommendation that

359

ROBUST CURRENCY MANAGEMENT

needs to be implemented at time t in order to shift the holding from si;t21 to si;t . We note that di;t is generally smaller in magnitude than si;t21 . During times when si;t21 implies holding a long position in a particular currency, a di;t of similar sign to si;t21 suggests an increase in the long position, whereas a di;t of opposite sign to si;t21 suggests a decrease in the long position. In the generic currency model, si;t21 is the main source of profit (loss), while di;t augments or mitigates the profit (loss), depending on its direction relative to si;t21 . The generic currency model Mi , through its time series of signals si;t , generates currency returns for the currency manager. Positive returns accumulate during periods when the model is forecasting correctly. Similarly, negative returns accumulate when the model mis-forecasts. It is during periods when mis-forecasts dominate that we would wish to intervene to adapt the management of currency portfolios that attempts to minimize the accumulation of losses to these portfolios. We present the minimax framework for this purpose in Section 5. 5 THE MINIMAX FRAMEWORK

5.1 Single Currency Framework We assume that the generic currency model presented in Section 4 results in a reasonable overall positive return for the currency manager. He/she continues to trade on the basis of the recommendations from the model, but recognizes the risk of incurring negative returns in following it. The return ri;t from holding a long position (see CN 3) in a particular currency i at time t, expressed as e , from a usd-based perspective, is given by 5 a function of ri;t e e ri;t ðri;t Þ ¼ si;t21 ri;t

e ri;t

¼

ei;t ei;t21

! 21

e where ri;t is the raw currency return due to a shift in the spot exchange rate, and ei;t is the exchange rate for currency i against the US dollar, quoted as units of US dollar per unit of currency i, at time t. For a model that produces daily signals, ri;t represents the daily currency return for holding the particular position si;t21 in currency i from the previous day. On a forward-looking basis, the potential return, ri;t11 , based on both the previous day’s signal si;t21 and the current day’s trade recommendation di;t , is given by 5

For ease of exposition, we concentrate on the currency return that excludes the forward bias. The use of the forward market in currency management introduces a forward bias in the currency return; this bias is generally small compared to the currency return based on spot values.

360

CHAPTER 11 e e ri;t11 ðri;t11 Þ ¼ ðsi;t21 1 di;t Þri;t11

e ri;t11

Eðei;t11 Þ 21 ei;t

¼

!

where the source of uncertainty is ei;t11 and EðzÞ denotes the expectations operator. In the multicurrency context, the return from all the currency exposures in the portfolio, rtP [ R1 , at time t is given by X e ri;t ðri;t Þ rtP ¼ i P

and the cumulative return r for a period of time is given by X P rP ¼ rt : t

This is the accumulated portfolio return over time which is important in assessing the overall health of a currency portfolio. Similarly, on a forwardlooking basis, the potential portfolio return, ri;t11 , at time t based on both the previous day’s signal si;t21 and the current day’s trade recommendation di;t , for all currency i is given by X P e rt11 ¼ ri;t11 ðri;t11 Þ: i P . The potential cumulative return is therefore the sum of r P and rt11 A manager can attempt to improve overall performance by concentrating on the component currencies that contribute to the overall return. We concentrate initially on single currencies and later on multicurrencies. For a particular currency i, the risk of a negative return on any day is generally compensated by the overall positive return over a number of days that the currency manager can get by following the model’s recommendations. In the context of the generic currency model described in Section 4, we assume that the horizon of the model is between 3 and 6 months. This means that for the currency manager, an overall positive return is expected in any 3–6 month period, but the risk of high negative returns within the period remains. The manager would attempt to minimize the occurrence of negative returns by considering the current performance of that currency component in the portfolio and the sign of the trade recommendation di;t relative to the signal si;t21 . The time series of si;t determines the performance over a period of the currency component within the portfolio. It is thus important to assess the accumulation of returns, whether positive or negative, in order to judge whether di;t , with a particular sign or direction relative to si;t21 , is a favorable trade, or not. Both si;t21 and di;t are component outputs of the generic currency model. Thus, it is assumed that the manager is not able to ignore di;t , mainly because it is a

361

ROBUST CURRENCY MANAGEMENT

necessary variable for shifting the signal from si;t21 to si;t . However, the manager can implement an overriding trade that would mitigate any potential increase in cumulative loss as a consequence of either si;t21 or di;t , or both. Let zi;t represent such an overriding trade, referred to as an overlay trade recommendation, that aims to minimize any potential cumulative loss. Consider the problem 8000 12 9 1 12 t < X  e  = e min @@@ ri;j A 1 ri;t11 ðri;t11 ÞA 1 zi;t ri;t11 A ð5:1Þ ; zi;t [R1 : j¼t 0

subject to

zlower # zi;t # zupper : i;t i;t In (5.1), ðzÞ2 ¼ minðz; 0Þ, t0 is some predefined starting time for the calculation of cumulative P&L (profit and loss). The constraint represents size restrictions on zi;t . This formulation attempts to minimize the potential cumulative loss in view of both si;t21 and di;t , as well as the expected move in the currency spot exchange rate Eðei;t11 Þ. In (5.1), the first term is the cumulative P&L with negative values only, that is, the running loss. If the cumulative P&L is positive, that is, the portfolio is accumulating profits, then the solution is zi;t ¼ 0. Hence the overlay trade does not interfere with the current performance of the generic currency model. However, if the cumulative P&L is negative, that is, the portfolio is accumulating losses, then the solution is a zi;t – 0, which would be positive if the expected return on the currency is positive and would be negative is the expected return on the currency is negative. The formulation does not require a budget constraint because this is not applicable in currency overlay. The upper and lower bound constraints on zi;t are chosen such that the buying or selling of a currency is within any guidelines on hedging imposed by the investor or self-imposed by the fund manager. The allocation to different currencies does not necessarily add up to some predefined budget, as such a budget is not relevant. The formulation given in (5.1) has a trivial solution for a single currency problem. If the first term is nonzero, then the optimizer will seek the upper limit or the lower limit on zi;t , depending on the expected currency return. This basic formulation is presented to illustrate the purpose of the overlay trade. In later formulations, we extend (5.1) to cover the multicurrency problem. It is important to note that the model trade recommendation di;t implies a  denotes the implied expected  i;t11 Þ, where EðzÞ currency spot exchange rate Eðe  value. A discrepancy between Eðei;t11 Þ and Eðei;t11 Þ may not necessarily result in a nonzero value for the overlay trade zi;t . If there is such a discrepancy, and if such discrepancy means opposite currency expectations, then the overlay e  i;t11 Þ within ri;t11 ðri;t11 Þ trade would only be activated if the contribution of Eðe is such that the running cumulative P&L results in a negative value. So long as

362

CHAPTER 11

the running cumulative P&L is positive, then the formulation will not intrude in the default currency management as indicated by si;t21 and di;t . A key criterion for the success of the above formulation would be the estimation of Eðei;t11 Þ. In order for this formulation to recognize an increasing P accumulation of loss due to the series si;t , the historic cumulative return, t ri;t , has to be part of the minimization process, and the overall sign of si;t during the period has to be analyzed in terms of its implied currency return. It may be prudent for the manager to set Eðei;t11 Þ equal to the implied currency return from si;t when si;t is producing positive returns, and similarly, to set it opposite the implied currency return from si;t when si;t is producing negative returns. The limitation of the above formulation is on the dependency of the solution on the estimation of Eðei;t11 Þ which may be driven by technical movements in the currency rather than being a reflection of a basic mis-forecasting by the model. This limitation can be addressed by the minimax formulation below, where we move away from an estimation of Eðei;t11 Þ to an estimation of a feasible range for ei;t11 . In the minimax framework, we wish to minimize the maximum potential cumulative loss due to the model recommendations. Consider the following problem related to currency i: 8000 1 112 0 t < X e 2 e i;t AA @@@ ri;j A 1 ðsi;t21 1 di;t Þ@ i;t11 min max e zi;t [R1 ei;t 1 1 [R1 : i;t j¼t0 1112 9 = e 2 e i;t AAA 1@zi;t @ i;t11 ; ei;t 0

0

ð5:2Þ

subject to upper elower i;t11 # ei;t11 # ei;t11

zlower # zi;t # zupper i;t i;t that is, upper and lower bounds on ei;t11 and zi;t . The estimation of the upper and lower bounds on ei;t11 , as in the estimation of Eðei;t11 Þ, depends on the implied currency returns from series si;t . However, whereas Eðei;t11 Þ is a point estimate, the upper and lower bounds produce a range estimate. The possibility of refining the estimation of this range provides a flexible formulation.

5.2 Single Currency Framework with Transaction Costs We present the equivalent formulations of (5.1) and (5.2) in view of transaction costs. These formulations, although simple in the single currency framework, lead to complications in the multicurrency framework.

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ROBUST CURRENCY MANAGEMENT

Problem (5.1) with transaction costs is given by (see CN 5) 8000 12 9 12 1 t t <   = X  e  X   e ki di;j 1 zi;j A min @@@ ri;j A 1 ri;t11 ðri;t11 ÞA 1 zi;t ri;t11 2 ; zi;t [R1 : j¼t j¼t 0

0

ð5:3Þ subject to

zlower # zi;t # zupper i;t i;t where ki . 0, that is, the cost 6 of transacting in currency i. The last term refers to the cumulative transaction cost from t0 to t. This term has a negative sign to indicate that it is a cost. While the costs due to di;z do not affect the objective function, its inclusion provides a more complete picture of the accumulation     of costs that may impact the cumulative P&L. Thus, the term di;j 1 zi;j  is constant in (5.3) up to j ¼ t 2 1 and is only optimized for j ¼ t. Problem (5.2) with transaction costs is given by (see CN 5) 8000 0 1 112 t < X e 2 e i;t11 i;t @@@ AA ri;j A 1 ðsi;t21 1 di;t Þ@ min max ei;t zi;t [R1 ei;t 1 1 [R1 : j¼t0 1 9 11   2= t X   e 2 e i;t11 i;t AA 2 1@zi;t @ ki di;j 1 zi;j A ; ei;t j¼t0 0

0

ð5:4Þ

subject to upper elower i;t11 # ei;t11 # ei;t11

zlower # zi;t # zupper : i;t i;t

5.3 Multicurrency Framework We present the multicurrency equivalent of (5.1), and show that certain cur complications arise when considering multiple currencies. Let Zt [ Rm be a vector whose elements zi;t represent a manager’s overlay trade recommendations in mcur currencies at time t. The multicurrency equivalent of (5.1) is given by 800 00 12 9 1 112 t = < X X X   e e A ð5:5Þ mincur @@ @@ ri;j A 1 ri;t11 ðri;t11 ÞAA 1 zi;t ri;t11 ; Zt [Rm : j¼t i i 0

6

Transaction costs depend on the currency traded and the volume of trade, and they could vary from 10 basis points to 100 basis points, where a basis point is a 100th of a percentage point.

364

CHAPTER 11

subject to

zlower # zi;t # zupper ; i;t i;t and where

2 6 6 Zt ¼ 6 6 4

i ¼ 1; …; mcur 3

z1;t

7 7 7: 7 5

.. .

zmcur ;t By expanding the components of the objective function, we have 800 00 0 1 1112 t < X X Eðe Þ 2 e i;t11 i;t AAA mincur @@ @@ ri;j A 1 ðsi;t21 1 di;t Þ@ ei;t Zt [Rm : i j¼t 0

1

X i

1112 9 = Eðe Þ 2 e i;t11 i;t AAA @zi;t @ ; ei;t 0

0

ð5:6Þ

subject to

zlower # zi;t # zupper ; i;t i;t and where

2 6 6 Zt ¼ 6 6 4

i ¼ 1; …; mcur 3

z1;t

7 7 7: 7 5

.. .

zmcur ;t Similarly, the multicurrency formulation of (5.2) is given by 800 00 1 0 10 1112 t < X X e 2 e i;t11 i;t AAA mincur max cur @@ @@ ri;j A 1 @si;t21 1 di;t A@ ei;t e^t 1 1 [Rm : Zt [Rm i j¼t 0

1

X i

1112 9 = e 2 e i;t11 i;t @zi;t @ AAA ; ei;t 0

0

subject to upper e^lower t11 # e^t11 # e^t11 upper elower ij;t11 # eij;t11 # eij;t11 ;

i ¼ 1; …; mcur 2 1

ð5:7Þ

365

ROBUST CURRENCY MANAGEMENT

j ¼ 1; …; mcur 2 1; ei;t11 2 e^t11

6 6 ¼6 6 4

e1;t11 .. .

1 ej;t11

3 7 7 7; 7 5

! 

 eij;t11 ¼ 1;

2 e^lower t11

i – j ðsee CN 4Þ

6 6 ¼6 6 4

elower 1;t11 .. .

3 7 7 7; 7 5

2 e^upper t11

6 6 ¼6 6 4

elower mcur ;t11

emcur ;t11

zlower # zi;t # zupper ; i;t i;t where

i–j

2 6 6 Zt ¼ 6 6 4

eupper 1;t11 .. .

3 7 7 7 7 5

eupper mcur ;t11

i ¼ 1; …; mcur

z1;t .. .

3 7 7 7: 7 5

zmcur ;t The second and third constraints refer to the triangulation requirement within a multicurrency framework. It is important to note that for the tactical currency minimax formulations, we do not use any currency model to define the upper and lower bounds on the currencies. Neither do we use a long-term currency model to define a cross exchange rate to satisfy triangulation requirements of currencies. In the tactical management of currencies using minimax, all we need are predefined upper and lower bounds on the currency, and an explicit triangulation requirement in terms of future exchange rates. The second and third constraints limit the choice of the maximizing variable to within reasonable values. Formulations (5.6) and (5.7) are the simple multicurrency expansions of (5.1) and (5.2). However, these may result in massive transaction costs because they do not account for cross hedging between currencies that do not involve transactions via the base currency. It is therefore important to include transaction costs in the formulation.

5.4 Multicurrency Framework with Transaction Costs Consider the equivalent of (5.6) when transaction costs are included (see CN 5):

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1 80 0 00 !112 12 9 t > > X X   > Eðe Þ 2 e i;t11 i;t > >B @ @@ r A 1 s AA C > > > i;j i;t21 1 di;t > > C B > ei;t =

j¼t0 C B mincur B C 0 1 ! ! B >   C Zt [Rm > t > > X X B X   C > Eðei;t11 Þ 2 ei;t > > A > @1 @   A > > 2 z k d 1 z > > i;t i i;j i;j   ; : e i;t i i j¼t0 ð5:8Þ subject to the same constraints as (5.6) and with K [ R of transaction costs, that is, 3 2 k1 7 6 6 . 7 K ¼ 6 .. 7: 5 4

mcur

denoting the vector

kmcur Similarly, consider the equivalent of (5.7) when transaction costs are included (see CN 5): 1 80 0 00 !112 12 9 t > > X X   > e 2 e i;t11 i;t > >B @ @@ r A 1 s AA C > > 1 d i;j i;t21 i;t > > C > B > ei;t =

j¼t0 C B mincur max cur B C 0 1 !! B >   C Zt [Rm e^t 1 1 [Rm > t > > X X B X   C > > ei;t11 2 ei;t > > A @1 @   A > > 2 zi;t ki di;j 1 zi;j  > > ; : ei;t i

i

j¼t0

ð5:9Þ subject to the same constraints as (5.7) and with the transaction cost vector cur K [ Rm defined as in (5.8). As transaction costs are summed for all currencies without any consideration for cross hedging capabilities, the currency manager could potentially buy and sell US dollars, and incur transaction costs, when there is no need for such transactions. He/she can reduce the number of transactions by taking advantage of any existing cross hedging opportunities. To illustrate the benefit from cross hedging, consider two foreign currencies A and B, for which the currency model recommends a buy trade of currency A accompanied by a sell trade of the US dollar, as well as a sell trade of currency B accompanied by a buy trade of the US dollar. By following the model’s trade recommendations, the currency manager incurs transaction costs on two US dollar trades, when one transaction involving a buy trade of currency A accompanied by a sell trade of currency B would yield the same result but with less cost.

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ROBUST CURRENCY MANAGEMENT

5.5 Worst-case Scenario The minimax formulation given in (5.9) is subject to constraints on the values that the future exchange rates may take. These constraints define the worstcase scenario within which the currencies in the portfolio take on future values that would result in the worst cumulative loss for the portfolio. In the context of the generic currency model described in Section 4, the signals and trade recommendations for each currency imply a distribution of future currency returns accompanied by an implied distribution of future values of the currency. The expected currency return and the dispersion about this expected value drives the model to generate a signal of a particular direction and magnitude. By extracting the distribution of future values of the currencies from the currency models, one can define the upper and lower bounds for the constraints in (5.9). To illustrate these constraints, let us assume that the currency model for yen generates a signal to hold a positive yen position, and a trade recommendation to buy more yen. Because the signal and the trade recommendation have the same direction, the currency model is effectively proposing an increase in the yen holding. This further implies that the yen is expected to appreciate within the horizon of the model. Depending on the success rate of the model in forecasting the movement of the yen, the distribution about the expected yen appreciation may vary. If the model results in low forecast errors, then the dispersion about the expected value may be tight; similarly, if the model results in high forecast errors, then the dispersion may be wide. The estimated mean myen;t11 and standard deviation syen;t11 of the future values of the yen can be used to define the upper bound and lower bounds on the yen, both defined by two standard deviations from the mean: upper elower yen;t11 # eyen;t11 # eyen;t11

where elower yen;t11 ¼ myen;t11 2 2syen;t11

ð5:10Þ

eupper yen;t11 ¼ myen;t11 1 2syen;t11 : By repeating this process for all currencies in the model, one is able to define the ranges of future values for each currency. As in Section 2, the triangulation property of currencies has to be preserved; all lower and upper bounds should satisfy triangulation. This is ensured by the second and third constraints in (5.7). The definition of the worst case can be refined further by considering the implied distributions of returns, and implied distributions of future values of exchange rates, for each factor in the model. In the generic currency model, the ultimate signal generated may be a result of an aggregation of signals from different factors that make up the model. To the extent that each of the factors

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within the model have varying forecasting capabilities throughout the model’s history, one can utilize the information contained within each factor to define the upper and lower bounds on the future values of the currencies. Assume that the currency model for our yen illustration generates a signal to hold a positive yen position which is dominated by two conflicting factors. A positive factor generates a signal to hold a positive yen position. A negative factor generates a signal in the opposite direction. The aggregation of the signals produced by each factor results in the overall model signal of holding a positive yen position. The constraints on the future values of the currencies, to be used in the minimax formulation, can be defined by considering any two dominant conflicting factors in the currency model. The estimated mean of the positive factor 1 myen;t11 of the future values of the yen can be used to define the upper bound on the yen, while the estimated mean of negative factor 2 myen;t11 can be used to define the lower bound. Consider the following bounding: upper elower yen;t11 # eyen;t11 # eyen;t11

where 2 elower yen;t11 ¼ myen;t11

ð5:11Þ

1 eupper yen;t11 ¼ myen;t11 :

Alternatively, one can extend the definition of the bounds by using the standard deviation of the positive factor, 1 syen;t11 , and the standard deviation of the negative factor, 2 syen;t11 , as upper elower yen;t11 # eyen;t11 # eyen;t11

where 2 2 elower yen;t11 ¼ ð myen;t11 Þ 2 ð syen;t11 Þ

ð5:12Þ

1 1 eupper yen;t11 ¼ ð myen;t11 Þ 1 ð syen;t11 Þ:

By defining the bounds as in (5.11) or (5.12), one recognizes that the conflicting factors within the model provide a clue as to the worst case. The positive factor that dominates the model signal would have a higher implied expected future value for the yen compared to that of the total signal and the negative factor that dominates the opposite direction. But as the negative factor is overwhelmed by the positive factor, it would have a lower implied future value compared to that of the total signal. It is the existence of a dominant opposing factor, in this illustration given by the negative factor, that may be backed by low forecasting errors that should be seriously considered when defining the worst case. Consider a situation where the negative factor in fact

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yields lower forecasting errors in the short term compared to the positive factor. If the overall model signal recommends a positive yen holding, while the negative factor within that model recommends a short holding of the yen, we would expect an accumulation of losses to the currency portfolio. By defining the lower bound based on the expected yen depreciation, as signaled by the negative factor, one is accounting for the conflicting nature of factors within the model. Repeating this process for all currencies in the model leads to the definition of the ranges of future values for each currency. As before, these ranges have to be refined further to ensure consistency in the sense that the triangulation property of currencies has to be preserved.

5.6 A Momentum-based Minimax Strategy The minimax formulation (5.9) can be further enhanced by considering the range of values of zi;t that would promote a more acceptable potential loss from the currency portfolio. In that formulation, the constraints on the future values of the currencies determine the worst case combination of exchange rates. By incorporating new constraints on the minimax overlay trade recommendations, one is promoting the search for the best case zi;t that would cushion the portfolio against the worst-case scenario. The growth or the accumulation of loss to the portfolio provides information about the suitability of the model’s trade recommendation di;t . A negative growth implies an increasing loss and an increasingly unfavorable forecast error, while a positive growth implies an increasingly healthy balance sheet and an increasingly favorable forecast error. The momentum of cumulative loss thus provides some information as to whether the model’s trade recommendation di;t should be implemented, or increasingly overlayed with an opposing zi;t . Consider the momentum-based minimax strategy given by (see CN 4): 1 80 0 00 !112 12 9 t > > X X   ei;t11 2 ei;t > > > AA C > B @ @@ ri;j A 1 si;t21 1 di;t > > > > C B > e i;t =

j¼t0 C B mincur max cur B C 0 1 !! > >B   C e^t 1 1 [Rm > Zt [Rm t > X X B X >   C ei;t11 2 ei;t > > A > @ @   A > > 2 1 z k d 1 z > > i;t i  i;j i;j  ; : e i;t i i j¼t0 ð5:13Þ subject to upper e^lower t11 # e^t11 # e^t11 upper elower ij;t11 # eij;t11 # eij;t11 ;

i ¼ 1; …; mcur 2 1

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j ¼ 1; …; mcur 2 1; ei;t11 2 e^t11

6 6 ¼6 6 4

e1;t11 .. .

1

! ðeij;t11 Þ ¼ 1;

ej;t11

3 7 7 7; 7 5

2 e^lower t11

i – j ðsee CN 4Þ

6 6 ¼6 6 4

elower 1;t11 .. .

i–j

3 7 7 7; 7 5

2 e^upper t11

6 6 ¼6 6 4

elower mcur ;t11

emcur ;t11

zlower # zi;t # zupper ; i;t i;t where

2 6 6 Zt ¼ 6 6 4

eupper 1;t11 .. .

3 7 7 7 7 5

eupper mcur ;t11

i ¼ 1; …; mcur

z1;t .. .

3 7 7 7 7 5

zmcur ;t and transaction cost vector K [ Rm Additionally, we have

cur

is as defined in (5.8).

  1 2 zupper ¼ f t ; t ; {r } i;t i i i;t

zlower ¼ 2zupper : i;t i;t The variables t1i and t2i are predefined time periods for estimating the momentum of cumulative loss and {ri;t }is the time series of currency returns. These t variables can be modeled separately from the currency model. The above formulation is also subject to triangulation constraints on the currencies. We now describe the function that defines the upper and lower bounds on zi;t . Between si;t21 and di;t , it is si;t21 that substantially contributes to the profit or loss of a portfolio. The cumulative profit or loss, likewise, is driven by {si;t }, that is, the time series of signals. A positive growth, or an accumulation of profit, suggests that, in the short term, {si;t } reflects a return distribution with a positive expected currency return, and perhaps accompanied by a small dispersion. Assuming that the short-term history of performance is an accumulation of profit, then an increasing accumulation, or the momentum of positive growth could itself be treated as a synthetic factor, separate from the factors that make up the generic currency model. In the case of an accumulation of loss, the momentum of negative growth could similarly be treated. Let t1i be a long time period, say 65 days, and let t2i be a short time period, say 5 days. One can

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approximate an increasing accumulation by taking the average return over these time periods and comparing the short-period average against the longperiod average. Let A1 and A2 be defined as 0 X

A1 ¼

t¼2 t1i

t1i

0 X

ri;t and

A2 ¼

t¼2 t2i

t2i

ri;t :

Here, A1 represents the average return for the long time period, while A2 is for the short time period. Then ( wðA2 2 A1 Þ if A2 $ A1 upper ð5:14Þ zi;t ¼ wðA1 2 A2 Þ if A1 . A2

zlower ¼ 2zupper i;t i;t where w is a weight that determines the size or magnitude of the signal based on some predefined rule. The modeling of the t variables to find the most appropriate time periods, as well as the modeling of the weight w, can be structured to be consistent with the generation of signals by the synthetic factor. The long-term performance of the synthetic factor should be comparable to the long-term performance of the other factors that make up the generic currency model. The importance of bounding zi;t as in (5.13) is that the aggressiveness 7 of such a decision variable is in line with the aggressiveness of the generic currency model. This makes for a tractable and systematic management of currency bets.

5.7 A Risk-controlled Minimax Strategy The minimax formulation in Section 5.6 attempts to control the risk of extreme deviations from the portfolio’s chosen benchmark by bounding the size of the overall trade recommendations. Formulation (5.9) can alternatively be risk-controlled by actively minimizing the tracking error that may result in following any trade recommendation. In (5.9), the constraints on the future values of the currencies determine the worst case combination of exchange rates. By incorporating a risk component into the formulation, one is promoting the search for the best case zi;t that would cushion the portfolio against the worst-case scenario, subject to an acceptable level of tracking error (TE). Consider the risk-controlled minimax strategy given by (see CN 4): 7

Aggressiveness is the term used to refer to the parameterization of a currency model signal that determines the size or magnitude of a currency bet that needs to be implemented in order to achieve a desired performance profile.

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1 80 0 00 !112 12 9 t   e > > X X > 2 e i;t11 i;t i > >B @ @@ r A 1 s AA C > > > i;j i;t21 1 dt > > C B > ei;t =

j¼t0 C B mincur max cur B C 0 1 ! ! B >   C e^t 1 1 [Rm > Zt [Rm t > > X X B X   C > ei;t11 2 ei;t > > A > @1 @   A > > 1 z k d 1 z > > i;t i i;j i;j   ; : e i;t i i j¼t0 ð5:15Þ subject to upper e^lower t11 # e^t11 # e^t11 upper elower ij;t11 # eij;t11 # eij;t11 ;

i ¼ 1; …; mcur 2 1;

j ¼ 1; …; mcur 2 1;

i – j ðsee CN 4Þ ei;t11 2 6 6 e^t11 ¼ 6 6 4

e1;t11 .. .

1 ej;t11

! ðeij;t11 Þ ¼ 1;

3

2

7 7 7; 7 5

6 6 6 e^lower ¼ t11 6 4

elower 1;t11 .. .

3

2

7 7 7; 7 5

6 6 6 e^upper ¼ 6 t11 4

elower mcur ;t11

emcur ;t11

zlower # zi;t # zupper ; i;t i;t where

i–j

2 6 6 Zt ¼ 6 6 4

eupper 1;t11 .. .

3 7 7 7 7 5

eupper mcur ;t11

i ¼ 1; …; mcur

z1;t .. .

3 7 7 7 7 5

zmcur ;t mcur

and transaction cost vector K [ R is as defined in (5.8). Additionally, a constraint on risk exposure is included:  t 1 Dt 1 Zt Þl # ðTEÞ2 kðSt 1 Dt 1 Zt Þ; CðS where TE [ R1 iscur ancuracceptable level of tracking error defined by the manager, C [ Rm x m is the covariance matrix of currency returns, and St 1 Dt 1 Zt is defined as

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2 6 6 ðSt 1 Dt 1 Zt Þ ¼ 6 6 4

s1;t 1 d1;t 1 z1;t .. .

3 7 7 7: 7 5

smcur ;t 1 dmcur ;t 1 zmcur ;t With the above formulation, the possibility of having a reasonable benchmark tracking performance is increased. If the realized future values of the exchange rates move outside their predefined limits used during optimization, then the constraint on tracking error would serve the purpose of controlling the volatility of the portfolio. 6 THE INTERPLAY BETWEEN THE STRATEGIC BENCHMARK AND TACTICAL MANAGEMENT Earlier sections of this chapter indicate the need for a tactical system in order for the portfolio to benefit from short- to medium-term fluctuations in currencies. In this section, we discuss the interplay between the strategic currency benchmark and the tactical currency trades. We consider the dominance of the strategic benchmark and the constraints it imposes on the tactical management of currencies. As in earlier sections, the term currency overlay is used in fund management to mean the application of the tactical currency trades to supplement and/ or complement the performance of the strategic currency benchmark. Because the strategic currency benchmark is designed to cover the risk of long-term depreciation in a currency, the corresponding currency hedge ratios drive the long-term performance of the benchmark. The strategic benchmark performance over the long term dominates the overall absolute performance of the portfolio currency exposure. In contrast, the excess performance provided by the tactical system is relatively small in proportion to the absolute performance of the strategic benchmark. Not only does the strategic benchmark impact the overall absolute performance, it also constrains the potential performance of the tactical system. The constraints imposed by the strategic benchmark limits the implementation of tactical trades. We illustrate the above constraint using a hypothetical yen exposure. Suppose that after the analysis of the strategic currency benchmark, the resulting hedge ratio for the yen exposure in an international portfolio is a 100% hedging recommendation. After the implementation of this hedge, the international portfolio essentially is devoid of any yen exposure. This total elimination of the yen exposure puts a constraint on the tactical system. This means that the tactical recommendation, either coming from di;t , orzi;t , or both, is restricted in terms of moving the yen exposure. If the tactical recommendation is a sell trade for the yen, then the tactical trade cannot be implemented

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because there is no longer any yen exposure to sell. Only buy trades can be accommodated by the tactical system for this fully hedged yen example. As another illustration, suppose that after the analysis of the strategic currency benchmark, the resulting hedge ratio for the yen exposure is a 50% hedging recommendation. After the implementation of this hedge, the international portfolio has essentially halved its yen exposure. Again, if the tactical recommendation is a sell trade for the yen, then the tactical trade can be implemented up to the remaining yen exposure, and not more. The constraint imposed by the strategic benchmark limits the potential performance of the tactical system, but does not fully eliminate it. In the implementation of currency overlay, currency managers have sought to regain some of the potential performance from their tactical systems by advocating some loosening of these constraints imposed by the strategic currency benchmark. In the illustration of the 100% hedging of the yen exposure, a loosening of the constraint may take the form of allowing net short positions in yen. In the illustration for the yen exposure, if the tactical recommendation is a sell trade for the yen, then the tactical trade can be implemented up to the allowable short yen exposure. The portfolio could then potentially benefit from the performance of this trade which would otherwise have been foregone if the constraint has not been loosened. 7 CURRENCY MANAGEMENT USING MINIMAX AND OPTIONS As with the use of options for managing asset portfolios from Chapter 9, the use of currency options for currency management is also attributed partly to their insurance capability. Currency managers in the import-export field are active users of currency options in view of the suitability of these for hedging expected cashflows from trade transactions. The periods within which these cashflows are expected to happen are short to medium term, and the use of options are deemed appropriate for providing the needed insurance policy for the term. However, the use of currency options by managers managing an international portfolio’s currency exposure takes a different form to that used by managers in the import-export field. In this case, options are held in the short and medium term up to the maturity of the option. Options are used as insurance providers, as well as being held for very short-term periods when they are used as return enhancers complementing tactical currency systems. Currency managers who provide overlay services for managing the currency exposure of international portfolios complement their overlay products by taking long or short positions in various currency options. The market provides a wide range of options: plain vanilla, asian, barrier, capped call, floored put, collar, lookback and quanto, to name a few. Additionally, combinations of options provide pay-out profiles that can be used for return

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375

enhancement. The reader is referred to Hull (1997) for a comprehensive discussion on these options. The generic currency model discussed in Section 4 and the tactical currency systems discussed in Section 5 do not preclude the use of options. However, a tactical formulation that includes options may not provide a practical solution for subscribers to currency overlay. This is due to the following three reasons. Firstly, a large majority of overlay subscribers would not allow the use of options. Those who would allow options tend to restrict their use to very specific conditions on the currency pairs, or on the type of options, or that positions should be long only, or that a very small currency exposure can be managed using options. Secondly, currency overlay managers attempt to diversify their product range by offering option-based currency management distinct from tactical currency systems. This prevents an active promotion of options within existing tactical systems. Lastly, data availability restrict the simulations that currency managers can do in searching for option-based strategies that may complement their existing tactical systems. This is an important restriction in trading systems development as well-defined excess return and tracking error profiles are essential in the world of benchmarkbased currency management. In Chapter 9, a minimax formulation that incorporates the use of options is presented as an enhanced portfolio management tool where insurance is provided by an optimal choice of out-of-the-money options. Such a framework cannot be adapted for tactical currency systems in Section 5 due to the very short-term nature of these systems. However, option-based currency overlay systems would have the ability to tailor options of varying horizons and they may be more amenable to minimax formulations. 8 CONCLUDING REMARKS In this chapter we discussed the need for currency management, mainly from the point of view of international portfolios where currency hedging is a critical issue in preserving the returns from the foreign assets that comprise the portfolio. We then subdivided the work of managing the currency exposure of an international portfolio in two ways: through a strategic currency management system that deals with the long-term direction of currencies, and through a tactical currency management system that deals with short-term fluctuations in particular currencies. The strategic currency management system identifies a long-term currency benchmark that provides the overall direction or bias of the currency hedge that needs to be implemented. The tactical currency management system identifies a short-term currency bet that improves on the already-implemented long-term currency benchmark. This short-term currency bet, as provided by a currency model signal, ensures that short-term currency fluctuations are

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utilized to the benefit of the portfolio. The excess returns that can be generated via a tactical currency management system supplement the returns that can be achieved from the strategic currency hedge. We presented minimax formulations for both the strategic and the tactical systems, and ways of identifying and evaluating worst-case scenarios. As currencies are constrained to move in relation to other currencies, the definition of worst-case scenarios are similarly constrained by the triangulation properties of exchange rates. References Eaker, M.R. and D.M. Grant (1990). ‘‘Currency Hedging Strategies for Internationally Diversified Equity portfolios’’, Journal of Portfolio Management, Fall, 30–32. Eun, C.S. and B.G. Resnick (1985). ‘‘Currency Factor in International Portfolio Diversification’’, Columbia Journal of World Business, Summer, 45–53. Eun, C.S. and B.G. Resnick (1988). ‘‘Exchange Rate Uncertainty, Forward Contracts and International Portfolio Selection’’, Journal of Finance, 43, 197–215. Hauser, S. and A. Levy (1991). ‘‘Optimal Forward Coverage of International Fixedincome Portfolios’’, Journal of Portfolio Management, Summer, 54–59. Hull, J. C. (1997). Options, Futures and Other Derivatives, Prentice Hall, London. Jorion, P. (1989). ‘‘Asset Allocation with Hedged and Unhedged Foreign Stocks and Bonds’’, Journal of Portfolio Management, 49–54. Levy, H. (1981). ‘‘Optimal Portfolio of Foreign Currencies with Borrowing and Lending’’, Journal of Money, Credit and Banking, 13, 325–341. Perold, A.F. and E.C. Schulman (1988). ‘‘The Free Lunch in Currency Hedging: Implications for Investment Policy and Performance Standards’’, Financial Analysts Journal, 45–50. Rosenberg, M.R. (1996). Currency Forecasting: A Guide to Fundamental and Technical Models of Exchange Rate Determination, Irwin, London. Rustem, B. (1995). ‘‘Computing Optimal Multicurrency Mean-variance Portfolios’’, Journal of Economic Dynamics and Control, 19, 901–908.

APPENDIX: CURRENCY FORECASTING Currency forecasting can be categorized into two major classes: fundamentalbased modeling and technical analysis. This appendix gives a brief overview of these models, following the comprehensive discussion in Rosenberg (1996). For further details on these models, the reader is encouraged to refer to Rosenberg (1996) and references therein. Forecasting models fall into two general categories: fundamental models and technical models. Associated with these are forecasting horizons that generally fall into three general categories: long-term forecasting where the emphasis is on structural and macro-economic forces that determine the equilibrium level of exchange rates, medium-term forecasting where an analysis of

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economic or business cycles may provide an insight into the cyclical position of exchange rates relative to the long-term equilibrium level, and short-term forecasting where the emphasis is on the analysis of speculative forces. While fundamental-based models appear to have a relative advantage in the mediumand long-term forecasting domains, technical models appear to have their relative advantage in the short-term domain. We give below a brief description of common fundamental models as well as technical models. Most fundamental models attempt to estimate the long-run equilibrium exchange rate level or path that the exchange rate will gravitate towards in the long run, and perhaps oscillate about in the medium run. In models based on purchasing power parity, it is assumed that nominal exchange rates would converge to a fair value that reflects differences in national inflation rates. In external balance-based models, it is assumed that nominal exchange rates would converge to a fair value that is consistent with the attainment of a balanced current account. Fundamental models that concentrate on the medium term fall in the general categories of asset-market models, monetary models, currency-substitution models and portfolio-balance models. In asset-market models of exchange rate determination, the supply of and demand for financial assets determine the medium-term trend that exchange rates take. In a monetary model, the supply of and demand for money determine the equilibrium exchange rate. In currency-substitution models, the anxiety of a nation in the local currency value erosion amplifies the volatility of the exchange rate and contributes to a perceived potential devaluation or depreciation in the currency. In the portfolio-balance models, the supply of and demand for money, as well as for bonds or government debt, determine exchange rate movements over medium-term periods. Fundamental models also consider the effect of economic variables such as interest rate differentials, fiscal policy changes and central bank intervention. Technical analysis has gained popularity due to its relative success in forecasting in the short term. However, it has been criticized as a long-term model. Despite this apparent shortcoming of technical analysis, market participants, particularly traders, use various models of technical analysis. These fall into two general categories: trend-following, where the model ascertains whether a trend is developing, and contrarian, where the model ascertains whether a trend is due for correction. Whether trend-following or contrarian, technical analysis can be subcategorized in terms of the technique used: charting, use of neural networks, signal processing and statistical or mathematical processes. Furthermore, within the domain of charting, a deeper categorization is possible in terms of the indicators produced by the charting analysis. These indicators generally fall under any of the following: moving average indicators, pattern recognition, oscillator indicators, divergence indicators, or trend indicators.

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The increasing trend in the use of technical analysis has been reinforced by the relative failure of fundamental models in generating short-term returns. However, market participants, particularly investors, realize that total reliance on a technical approach to currency forecasting can be very risky when false technical signals resulting from weak trending markets give rise to huge losses. Investors tend to avoid a strong reliance on technical signals especially when fundamental signals do not support or reinforce those signals. Additionally, investors tend to look not only at the short term, where technical models are relatively more useful, but at the medium and long term as well, where fundamental models are relatively more useful. There is a need to address the balance between the use of technical and fundamental models in order for market participants to minimize the risk of incurring currency losses due to mis-forecasting. Indeed, there has been a tendency to base a long-term currency view on fundamental models and a tendency to base a short-term currency view on technical models, and a tendency to weight any aggregation of signals are on the basis of the relative importance of making a long-term view as opposed to a short-term view. COMMENTS AND NOTES

CN 1: Hedging of Currency Risk Hedging is the technical term in finance to refer to the implementation of a strategy to mitigate any potential unfavorable outcome from holding a position. In the context of holding a currency portfolio or an international portfolio with currency exposures, hedging refers to the strategy of eliminating all or part of the potential negative return if a currency moves against the investor. The concept of base currency is very important in ascertaining the appropriate hedge. For a usd-based investor who invests in a foreign country’s equity market, the currency risk comes from having to translate the gains (or losses) from the equity market into equivalent gains (or losses) in US dollar terms. If the foreign currency depreciates relative to the base currency, then the equivalent gain (or losses) in US dollar terms gets eroded. Generally, the hedging of currency risk involves the use of a forward currency contract that stipulates the exchange rate to apply to a particular nominal amount of the foreign currency for exchange back to the base currency at a future time. In complex hedging strategies, forward, swap, option, and spot transactions may be employed.

CN 2: Cross Hedging A cross hedge refers to the implementation of a currency hedge when the currencies involved do not include the base currency. Any potential depreciation of the first currency relative to the second currency is mitigated by selling

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the first currency and buying the second currency. A cross hedge does not necessarily mean an improvement in the overall risk exposure of a portfolio from the point of view of the base currency. The reason for this lack of certainty is that a cross hedge has to depend on the movement of the bought currency, in this case the second currency, relative to the base currency.

CN 3: Long Position versus Short Position in a Currency The terms ‘‘long’’ and ‘‘short’’ a currency refer to the holding of a foreign currency. A long position means that the investor owns the currency; this currency may physically reside in a deposit account or it may be invested in an asset denominated in that currency. A short position means that the investor does not own the currency but has sold the currency. This is possible in a situation where the investor enters into a forward contract to sell the currency even if she does not physically have notes and coins, or assets to back the currency.

CN 4: Cross-currency Constraints The number of cross-currency constraints is given by the combination of ðmcur 2 1Þ currencies taken two at a time, that is, ðmcur 21Þ

C2 ¼

ðmcur 2 1Þ! : ððmcur 2 1Þ 2 2Þ!2!

CN 5: Implementing the Transaction Cost Term

  The overall transaction cost depends on the magnitude of di;t 1 zi;t . This term can be incorporated within the setting of the quadratic programming formulation using a simple reformulation. We note that in the formulations where a transaction cost term appear, only the variable part, zi;t , is considered. Let 2 2 di;t 1 zi;t ¼ x1 with x1 i;t 2 xi;t ; i;t ; xi;t $ 0   2 1 2 di;t 1 zi;t  ¼ x1 i;t 1 xi;t ; and xi;t ¼ 0 if xi;t . 0 1 and x2 i;t ¼ 0 if xi;t . 0:

Thus, the transaction cost component X   2 ki di;t 1 zi;t  t

is replaced in the objective by

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X

 2 1 2 2 ki x1 i;t 1 xi;t 1 cxi;t xi;t

t

with added constraints 2 di;t 1 zi;t ¼ x1 i;t 2 xi;t

and

2 x1 i;t ; xi;t $ 0:

2 We assume that c . 0 is chosen to be sufficiently large to ensure x1 i;t £ xi;t ¼ 0.

Index abrupt change variants 196–8, 201–3, 222–3 accuracy 112–18 actual hedging errors 192–5, 212–13 aggressiveness 371 ALB see Asset/Liability Management American bond options 229–33 American call options 182 ancestor states 279, 331 approximate Jacobians 55–8 Armijo stepsize strategy 50–8, 82–3 asset allocation 247–90 benchmarking 247–52, 261–71 downside risk 271–3 dual benchmark tracking 261–71 forecasting 273–7 multistage minimax bond portfolios 277–84 rival return forecasts 249–52 threshold returns 271–3 Asset/Liability Management (ALM) continuous minimax 295, 308–9 immunization 292–303 multivariate immunization 295–303 risk in immunization 303–7 stochastic models 315–34 uncertainty 291–340 univariate convexity 312–15 univariate duration 309–12 associated risk 250 ‘‘at-the-money’’ 189 augmented Lagrangians 132–7, 142

backtesting 258–60 balance sheet restructuring 318–19 barbell strategies 293 barrier functions 46–9 benchmarking asset allocation 247–52, 261–71, 290 currency management 341, 345–80 beta of the hedge 226 BFGS see Broyden-FletcherGoldfarb-Shanno binomial trees 229–33, 315–18 Black and Scholes (BS) 183–7, 244–5, 287–8 bonds liabilities 300–3 management 252–61 minimax 252–60, 277–84 options 226–33 portfolios 252–61, 264–71, 277–84 prices 231 boundaries, downside risk 274–7 bounded sets 20–1 British Telecom 239, 240–1, 242, 243 Brownian Motion 244 Broyden update 50 Broyden-Fletcher-Goldfarb-Shanno (BFGS) formula 72–5 BS see Black and Scholes bullet strategies 293 bundle nonsmooth optimization 23–5

382 Cadbury 240 call options 181–2, 204–6, 214 Capital Asset Pricing Model (CAPM) beta of the hedge 226 continuous minimax 179 minimax hedging 215–22, 234 simulation study 222–5 Caratheodory’s theorem 7–9 cash-matching 292–5, 303–4, 323 cashflow 191, 321 Chaney’s Method 24, 25–6, 27–8 closed sets 20–1, 60 combination strategies currency management 374–5 portfolio management 284–8 compact sets 20–1, 60, 63 concavity 238–44 constraints asset allocation 248 continuous minimax 25–6 cross-currency 379 discrete minimax 99, 108–10, 139–78 saddle points 38, 44–9 contingent assets/liabilities 319–20 continuous functions 36 continuous minimax Asset/Liability Management 295, 308–9 delta hedging 180–7, 194 hedging 179–245 introduction 2, 10–11 numerical experiments 93–120 options hedging 179–245 quasi-Newton algorithm 63–92 survey 23–35 convergence 50–61 discrete minimax 152–6 Newton-type algorithms 81–6 quasi-Newton algorithm 76–81 sequences 20–1 convexity 3–5

INDEX

Asset/Liability Management 309–15, 337–8 concave continuous minimax 98–103 continuous minimax 98–103 discrete minimax 132–7 quasi-Newton algorithm 63 subdifferential functions 7–9 subgradient functions 7–9 cross exchange rates 350 cross hedging 366, 378–9 cross-currency constraints 379 crossovers 196–8 cumulative normal distribution function 244–5 cumulative returns 260, 270 currency forecasting 342, 356, 376–8 currency hedge ratio 353–7 currency hedging 341–80 currency management 341–80 minimax 359–75 momentum-based minimax strategy 369–71 risk-controlled strategy minimax strategy 371–3 currency mean-variance system 343 currency overlay 351–7, 361 currency risk 378 dk 75–6, 149 Datastream International 205 date of strategy shift 317 debts 342 dedication see cash-matching ‘‘deep-in-the-money’’ 201 ‘‘deep-out-of-the-money’’ 201 definite matrices 19 degree of moneyness 201 delta hedging Capital Asset Pricing Model 222–5 continuous minimax 180–7, 194 minimax strategy 206 performance 199–201

INDEX

Dennis-More´ characterization 56–8, 83–4 descent dk property of 75–6, 149 evaluation 97–8 motivation 34–5 quasi-Newton algorithm 90–1 differentiable functions 19, 36 direction 69–71, 90–2, 97–8 directional derivatives 7 directional immunization 298–303, 308–9 discount bonds 229 discount rates 205 discrete minimax 2 American bond options 231 Asset/Liability Management 295 augmented Lagrangians 132–7 convergence 152–6 convexity 132–7 equality constraints 139–78 global convergence 152–6 guaranteed performance 126–8 inequality constraints 139–78 linear constraints 172–6 noninferiority 126–8 quadratic programming 143–4, 162–72 robustness 121–38 superlinear convergence 162–72 unit stepsize strategy 144–5, 156–62 dividends 198 dollar bond portfolios 253–6, 261, 264–71 convexity 309–15 duration 309–15 exchange rates 345–51, 358 investments 343 London Inter-Bank Offered Rate 261, 264–71 yen rates 345–51, 358

383 downside risk 248, 271–3, 290, 323 dual benchmark tracking 261–71 dual-optimal bond portfolios 266–71 duration 309–15, 337–8 dynamic hedging strategy 182 dynamic multistage stochastic Asset/ Liability Management 325–9 economic forecasting 279 efficiency 252 endowment funds 271 epigraphs 7 equality constraints 139–78 equilibria of saddle points 37–40 equity 341, 352–7 equivalence of direction 69–71 error variables 221 Euclidian norm 19 European bond options 226–9 European call options 181–2, 204–6, 214 European put options 287–8 exact Jacobians 54–5 exchange rates 345–51, 361, 367–9 exchange traded options 189 exercise price 181, 189, 197–201 extreme point solution 239, 240–1, 243 finiteness 150–2 Finsler’s Lemma 133–5 first order Taylor expansions 19–20 fixed returns 251 fixed risks 250 fluctuation management 357–9 forecasting asset allocation 249–60, 273–7, 279 currency management 342, 376–8 discrete minimax 173–4 rival decision models 121–3 foreign assets 342 foreign-denominated debts 342

384 frontiers in asset allocation 255–60, 266–71 full hedging 344 generic currency management model 357–9 global benchmark tracking 264–6, 267–71 global convergence 50–8, 81–6, 152–6 global minima 22 gradient-based algorithms 42–3 guaranteed performance 11, 126–8 Guinness 240–2, 243 Haar condition 11, 13–15 hedge ratios 352–7 hedging American bond options 229–33 asset allocation 284–8 bond options 226–33 continuous minimax 179–245 credit 295 currency risk 341–80 errors 192–5, 211–13, 228–9 European bond options 226–9 synthetic assets 354–7 two-period minimax 207–15 variable minimax 207, 211–15 Hessian 65, 72–5, 89–90 high performing variants 202–4 horizons 307, 316–17, 343, 346 hyperplanes 21 immunization 292–315 implementation issues Kiwiel’s algorithm 96–7 quasi-Newton algorithm 87–90, 96–7 implied volatility 205 ‘‘in-the-money’’ 188, 229 index tracking see benchmarking inequality constraints 139–78

INDEX

inner product evaluation 18–19, 68 interest rates 205, 229, 295–6, 338 interior point algorithm 45–9 international bonds 255–6 international currency management 341–80 Ito’s Lemma 244 j-step Q-superlinear rate 61 Jacobians 41, 49–50, 54–8 Karush-Kuhn-Tucker conditions 35–6 kinks 65 Kiwiel’s algorithm 24–5, 31–3 accuracy 112–18 implementation 96–7 max-function 94–7 stopping criterion 97 superlinear convergence 111–12 termination criterion 112–18 terminology 96–7 Lagrangians 132–7, 142, 148–9 level variants 222, 223 liability see Asset/Liability Management LIBOR see London Inter-Bank Offered Rate linear constraints 172–6 linear independence 5–7, 20 Lipschitz continuity 21 local asset returns 353 local convergence 81–6 local minima 22 local Q-superlinear convergence rate 83–4 local superlinear convergence rate 56–8 London Inter-Bank Offered Rate (LIBOR) 261, 264–71 long positions 379 lower bounds, downside risk 274–7

INDEX

Macaulay Duration 337–8 management asset/liabilities 291–340 bonds 252–61 currency 341–80 market index movements 221, 222 Market Model 218 market-capitalization-weighted global benchmark 264–6 Markowitz frameworks 253–4 matrices 18–19 max-function Hessian 65 introduction 2–5 Kiwiel’s algorithm 94–7 monotonic decrease 76–81 maximizers 10, 71–3, 89–90, 238–44 mean value theorem 59 mean-variance asset allocation 273–7 currency management 343 optimization 173, 247–8, 253–4 mid-range solutions 240, 241–3 minima 22 minimax asset allocation 247–90 bond portfolios 252–61, 277–84 combination currency management 374–5 combination portfolio management 284–8 currency management 359–75 high performing variants 202–4 introduction 1–22 multicurrency management 363–6 naive 125–6, 132, 174 robustness 11–15, 195 saddle points 44–5 single currency management 359–63 stochastic Asset/Liability Management 330–5 tactical currency management 365

385 minimax hedging 179–245 beta of the hedge 226 Capital Asset Pricing Model 215–22 errors 183, 189–90, 192–3 European call options 204–6 multiperiods 207–13 simulations 196–204 variants 194 minimum-norm subgradient 111 models asset allocation 271–3 Asset/Liability Management 315–34 Black and Scholes 183–7 Capital Asset Pricing 215–22 currency management 357–9 forecasting 121–3, 376–8 Value-at-Risk 271–3 Modified Duration 337–8 momentum-based minimax currency management 369–71 moneyness 201 monotonic decrease 51–3, 76–81 mortgages 319 multicurrency management 343–80 multidimensional immunization 295–303 multiperiod minimax 207–15, 234 multiple maximizers 89–90, 93 multistage Asset/Liability Management 325–33 multistage minimax bond portfolios 277–84 multivariate immunization 295–303 naive minimax 125–6, 132, 174 nce: necessary condition for an extremum see optimality conditions Newton algorithms direction 35 global convergence 50–3, 81–2

386 see also quasi-Newton algorithms no hedging 344 nodes 230, 278–9, 315–17 nonconvex-nonconcave continuous minimax 98 nonextreme point solutions 238–9 noninferiority 126–8 nonlinear quasi-Newton algorithm 49–50 nonnegativity 321 nonsatiation 185 nonsmooth optimization 23–5 normal cones 7 numerical examples, options hedging 237–43 numerical experiments, continuous minimax 93–120 objective functions American bond options 231–2 Capital Asset Pricing Model 219–21 discrete minimax 128–30 minimax hedging 190–2 two-period minimax 208–9 one-period trinomial trees 230–1 open ball 20, 60 open sets 20–1 optimal hedge ratios 352–7 optimality conditions 11–15, 35–6, 166 optimization 23–5 mean-variance 173, 247–8, 253–4 options 221 American bonds 229–33 American call 182 bonds 226–33 call 181–2, 204–6, 214 combination currency management 374–5 combination portfolio management 284–8 contracts 181

INDEX

European bonds 226–9 European call 181–2, 204–6, 214 European put 287–8 exchange traded 189 hedging 179–245 pricing 183–7 put 181, 287–8 order of o(x), O(x) 21 orthogonality 18–19 ‘‘out-of-the-money’’ 189, 284–8 overlay trade recommendation 361 Panin’s algorithm 24, 30–1 partial hedging 344 ‘‘payoff matrix’’ 121, 131 payout dates 316–17 penalty formulation 46 penalty parameters 142, 145–8, 150–2 pension fund management 271 Pironneau-Polak method of centres 26–7 pooling minimax formulation 123–5 pooling weights 122–3 portfolios asset allocation 247–90 bonds 252–61, 264–71, 277–84 combination management strategies 284–8 currency management 345–51 hedging 226 performance backtesting 258–60 pure currency 345–51 spot exchange rates 349 strategic currency management 345–51 positive definiteness 63 positive semi-definite matrices 19 potential hedging errors 190–2, 212–13, 231 preference independence 186 present allocation strategies 247–90 price determination functions 217–18 Prudential 240–1, 242, 243

INDEX

pure currency portfolios 345–51 put options 181, 287–8 Q-linear rate 61 Q-superlinear convergence rate 56–8, 61, 83–4, 170–2 quadratic approximation 44–5 quadratic programming 143–4, 162–72 quasi-Newton algorithms accuracy 112–18 concepts 66–70 continuous minimax 63–92 convergence 76–86, 111–12 definitions 66–70 implementation 87–90, 96–7 introduction 23–5 maximizers 91–2 nonlinear systems 49–50 numerical experiments 95–7 Q-superlinear convergence 56–8, 83–4 saddle points 49–50 stepsizes 82–3 stopping criterion 97 superlinear convergence 111–12 termination criterion 112–18 terminology 96–7 unconstrained saddle points 49–50 range forecasts 273–7 rank ordering 214–15, 224–5 rebalancing minimax hedging strategy 190 return trade-offs 255–60 risk controlled minimax strategy 371–3 risk free interest rates 205 risk in immunization 303–7 risk tolerance 248 risk trade-offs 255–60 rival forecasting 121–3, 173–4, 249–52

387 rival risk asset allocation 249–52 robust currency management 341–80 robust hedging strategies 179, 231 robustness of discrete minimax 121–38 robustness of minimax 11–15 saddle points algorithms 37–61 computation 37–61 conditions 15–16 equilibria 37–40 introduction 15–16 quasi-Newton algorithm 49–50 solving the system of equations 40–3 second order Taylor expansions 19–20 semi-definite matrices 19 sequence convergence 20–1, 61 setting up hedges 198, 204–5 short positions 379 short-term currency fluctuations 357–9 simplified quasi-Newton algorithm 95–7 simulation studies 196–204, 213–15, 222–5 single currency management 359–63 single-stage Asset/Liability Management 333–5 solving the system of equations 40–3 split-variable formulation 330–1 spot curves 303, 308 spot exchange rates 349, 361 standard deviations in hedging 189 states 315–17 static hedging strategy 182 stepsizes convergence 50–8 discrete minimax 144–5, 156–62 quasi-Newton algorithm 82–3

388 stochastic Asset/Liability Management 315–34 stock prices 184, 197–8 stopping criterion 97, 112–18, 237–8 strategic currency management 345–57 strict global minima 22 strict local minima 22 subdifferentials 7–9 subgradient convex functions 7–9 subgradient nonsmooth optimization 23–5 superlinear convergence 54–8, 111–12, 162–72 synthetic assets 354–7 tactical currency management 357–9, 365, 373–4 tangent cones 7 Taylor expansions 19–20 term structures 315–17, 338 terminal dates 316–17 terminal wealth 327 termination criterion 97, 112–18, 237–8 Tesco 240–1, 243 test functions 28–9 tests constrained discrete minimax 108–10 convex-concave continuous minimax 98–103 convex-convex continuous minimax 103–8 unconstrained continuous minimax 108–10 Thames Water 240–1, 243 threshold returns 271–3 tracking errors 261, 268 transaction costs American bond options 231 Black and Scholes option pricing 186

INDEX

Capital Asset Pricing Model 220 currency management 379–80 minimax hedging 192, 193–4, 201, 203 multicurrency management 365–6 single currency management 362–3 two-period minimax 210 Treasury Bill value 205 tree structures American bond options 229–33 Asset/Liability Management 315–17, 338–9 binomial 229–33, 315–17 multistage minimax bonds 278–84 trinomial 229–33 triangulation 350–1, 355 trinomial tree 229–33 two asset allocation strategies 254–6 two-period minimax 207–15, 234 uncertainty Asset/Liability Management 291–340 Capital Asset Pricing Model 221 unconstrained continuous minimax 99, 108–10 unconstrained saddle points 38, 42–3, 49–50 underlying stock 181 unique maximizers 89–90, 91–2 uniqueness condition 39–40 unit stepsizes convergence 50–8 discrete minimax 144–5, 156–62 quasi-Newton algorithm 82–3 univariate convexity 312–15 univariate duration 309–12 upper bounds, downside risk 275–7 US dollar see dollar

INDEX

Value-at-Risk models 271–3 variable minimax 207, 211–15, 234 vectors 18–20 volatility 196, 202–4, 216 weighted minimax variants 199–204, 236 weighting matrices 192, 210, 220

389 winding down hedges 198, 204–5 yen/dollar rates 345–51, 358, 368 yield curves American bond options 229–33 Asset/Liability Management 311–12, 318–19 pure currency 348